Lecture 15

class: center, middle, inverse, title-slide

# Lecture 15
## Factorial Designs
### Psych 10 C
### University of California, Irvine
### 05/04/2022

---

## Factorial Designs

- Last class we worked with an example of a `\(2\times2\)` between subjects 
factorial design using the cell means approach.

- Today we will work with a new example using R.

- First we will introduce a new study.

---

## Depression example

- Researchers are interested on the effects of a pharmacological treatment and 
therapy for the treatment of depression.

- They have design a study with 100 participants divided in 4 groups. The first
group did not receive any treatment (no pharmaceutical treatment and no 
therapy).

- The second group was exposed only to therapy for 1 month.

- The third group was only exposed to a pharmaceutical treatment for 1 month.

- The last group was exposed to both, a pharmaceutical treatment and therapy for 
1 month.

- The dependent variable in the study was the difference between pre/post 
treatment scores on a depression inventory (scale).

---

## Depression example

- This is a `\(2\times2\)` between subjects factorial design. The first factor is 
whether participants received pharmaceutical treatment or not. We denote
with `\(j = 1\)` participants that did not receive pharmaceutical treatment.

- The second factor is whether the participant was exposed to therapy or not.
We denote with `\(k = 1\)` participants that did not receive therapy.

- As we know, there are 5 models that we need to compare for this type of study.

- The Null model assumes that the difference in depression scores between 
pre/post treatment will be the same regardless of the group that a participant
belongs to (combination of the levels of our factors).

---

## Depression example

- A pharmaceutical treatment main effects model, which assumes that only 
pharmaceutical treatment has an effect on the difference between pre/post
treatment depression scores.

- A therapy main effects model, which assumes that only therapy has an effect on 
the difference between pre/post treatment depression scores.

- An additive model, which assumes that both pharmaceutical treatment and 
therapy have an effect on the difference between pre/post treatment depression
scores. The effects according to this model are independent.

- Finally, we have the full model which assumes that the effect of a 
pharmaceutical treatment on the difference between pre/post treatment 
depression scores of a participant depends on whether they where also exposed to 
therapy or not.

---

## Experimental data

- The first thing that we need to do is look at the data. Because we have two
independent variables (pharmaceutical treatment and therapy) a box-plot would 
be useful:

.pull-left[

```r
ggplot(data = depression) + 
  aes(x = therapy) + 
  aes(y = beck_diff) +
  aes(color = pharma_treatment) + 
  geom_boxplot(position = position_dodge(1), outlier.shape = NA) + 
  geom_dotplot(data = depression, binaxis = 'y', stackdir='center', 
               dotsize = 0.37, method = "histodot", position_dodge(1),
               alpha = 0.2, binwidth = 0.8,
               mapping = aes(x = therapy, y = beck_diff, color = pharma_treatment,
                             fill = pharma_treatment), show.legend = FALSE) +
  theme_classic() + 
  xlab("Therapy") + 
  ylab("Depression scores (pre - post)") + 
  guides(color = guide_legend("Pharma")) + 
  theme(axis.title.x = element_text(size = 25), 
        axis.title.y = element_text(size = 25),
        axis.text = element_text(size = 20))  
```
]
.pull-right[
<img src="data:image/png;base64,#lec-15_files/figure-html/hist-score-out-1.png" style="display: block; margin: auto;" />
]